Messoud Efendiev, Vitali Vougalter
Existence of solutions for some systems of integro-differential equations
with transport and superdiffusion
(295K, pdf)
ABSTRACT. We establish the existence in the sense of sequences of solutions for certain systems of integro-differential equations which involve the drift terms and the square root of the one dimensional negative Laplace operator, on the whole real line or on a finite interval with periodic boundary conditions in the corresponding H^2 spaces. The argument is based on the fixed point technique when the elliptic systems contain first order
differential operators with and without Fredholm property. It is proven that, under the reasonable technical conditions, the convergence in L^1
of the integral kernels yields the existence and convergence in H^2 of the solutions. We emphasize that the study of the systems is more complicated
than of the scalar case and requires to overcome more cumbersome technicalities.